Lie Bialgebra Research Articles

LIE BIALGEBRAS ARISING FROM POISSON BIALGEBRAS

It gives a method to obtain a natural Lie bialgebra from a Poisson bialgebra by an algebraic point of view. Let g be a coboundary Lie bialgebra associated to a Poission Lie group G. As an application, we obtain a Lie bialgebra from a sub-Poisson bialgebra of the restricted dual of the universal enveloping algebra U(g).

Quantization of lie algebras of block type

In this article, we use the general method of quantization by Drinfeld's twist to quantize explicitly the Lie bialgebra structures on Lie algebras of Block type.

New r-Matrices for Lie Bialgebra Structures over Polynomials

For a finite dimensional simple complex Lie algebra $${\mathfrak{g}}$$ , Lie bialgebra structures on $${\mathfrak{g}\left[\left[u \right]\right]}$$ and $${\mathfrak{g}\left[u\right]}$$ were classified by Montaner, Stolin and Zelmanov. In our paper, we provide an explicit algorithm to produce r-matrices which correspond to Lie bialgebra structures over polynomials.

Lie Bialgebra Structures on the Centerless W-Algebra W(2,2)

Verma modules over the W-algebra W(2,2) were considered by Zhang and Dong, while the Harish-Chandra modules and irreducible weight modules over this algebra were classified by Liu and Zhu etc. In the present paper, we investigate the Lie bialgebra structures on the centerless W-algebra, which are shown to be triangular coboundary.

A construction for coisotropic subalgebras of Lie bialgebras

Given a Lie bialgebra ( g , g ∗ ) , we present an explicit procedure to construct coisotropic subalgebras, i.e. Lie subalgebras of g whose annihilator is a Lie subalgebra of g ∗ . We write down families of examples for the case that g is a classical complex simple Lie algebra.

Quantization of coboundary Lie bialgebras

We show that any coboundary Lie bialgebra can be quantized. For this, we prove that: (a) Etingof-Kazhdan quantization functors are compatible with Lie bialgebra twists, and (b) if such a quantization functor corresponds to an even associator, then it is also compatible with the operation of taking coopposites. We also use the relation between the Etingof-Kazhdan construction of quantization functors and the alternative approach to this problem, which was established in a previous work.

Lie bialgebras arising from alternative and Jordan bialgebras

We consider connection between simple alternative D-bialgebras and Lie bialgebras. We prove that each finite-dimensional alternative noncommutative algebra over an algebraically closed field may be equipped with the nontrivial structure of an alternative D-bialgebra.

Nonabelian Generalized Lax Pairs, the Classical Yang-Baxter Equation and PostLie Algebras

We generalize the classical study of (generalized) Lax pairs and the related $O$-operators and the (modified) classical Yang-Baxter equation by introducing the concepts of nonabelian generalized Lax pairs, extended $\calo$-operators and the extended classical Yang-Baxter equation. We study in this context the nonabelian generalized $r$-matrix ansatz and the related double Lie algebra structures. Relationship between extended $O$-operators and the extended classical Yang-Baxter equation is established, especially for self-dual Lie algebras. This relationship allows us to obtain explicit description of the Manin triples for a new class of Lie bialgebras. Furthermore, we show that a natural structure of PostLie algebra is behind $O$-operators and fits in a setup of triple Lie algebra that produces self-dual nonabelian generalized Lax pairs.

Quantization of Poisson–Hopf stacks associated with group Lie bialgebras

Let G be a Poisson Lie group and g its Lie bialgebra. Suppose that g is a group Lie bialgebra. This means that there is an action of a discrete group Γ on G deforming the Poisson structure into coboundary equivalent ones. Starting from this we construct a non-trivial stack of Hopf-Poisson algebras and prove the existence of associated deformation quantizations. This non-trivial stack is a stack of functions on the formal Poisson group, dual of the starting Γ Poisson-Lie group. To quantize this non-trivial stack we use quantization of a Γ Lie bialgebra which is the infinitesimal of a Γ Poisson-Lie group (cf [MS] for simple Lie groups and Γ a covering of the Weyl group and [EH2] for quantization in the general case).

Quantization of quasi-Lie bialgebras

We construct quantization functors of quasi-Lie bialgebras. We establish a bijection between this set of quantization functors, modulo equivalence and twist equivalence, and the set of quantization functors of Lie bialgebras, modulo equivalence. This is based on the acyclicity of the kernels of natural morphisms between the universal versions of Lie algebra cohomology complexes for quasi-Lie and Lie bialgebras. The proof of this acyclicity consists of several steps, ending up in the acyclicity of a complex related to free Lie algebras, namely, the universal version of the Lie algebra cohomology complex of a Lie algebra in its enveloping algebra, viewed as the left regular module. Using the same arguments, we also prove the compatibility of quantization functors of quasi-Lie bialgebras with twists, which allows us to recover our earlier results on compatibility of quantization functors with twists in the case of Lie bialgebras.

Double cross biproduct and bi-cycle bicrossproduct Lie bialgebras

We construct double cross biproduct and bi-cycle bicrossproduct Lie bialgebras from braided Lie bialgebras. The main results generalize Majid's matched pair of Lie algebras and Drinfeld's quantum double and Masuoka's cross product Lie bialgebras.

Quantization of the q-analog Virasoro-like algebras

We use the general method of quantization by Drinfel'd twist element to quantize explicitly the Lie bialgebra structures on the q-analog Virasoro-like algebras studied in Comm. Algebra, 37 (2009), 1264{1274.

Lie Algebras with a Coalgebra Splitting

In their recent article [5], the authors endow every finite-dimensional simple complex Lie algebra g with a coalgebra structure such that the composition μ◦δ of the two structure maps δ : g −→ g⊗Cg and μ : g⊗C g −→ g coincides with the identity. Moreover, the dual algebra (g∗, δ∗) associated to the Lie coalgebra is isomorphic to (g, μ). The coalgebra map δ is given explicitly for sl(n), those for the other types are obtained via embeddings g ↪→ sl(n). The purpose of the present short note is to elicit the conceptual sources of [5], starting from the observation that the coalgebra maps defined in [5] are in fact homomorphisms of g-modules. For Lie algebras affording non-degenerate symmetric associative forms, such coalgebra maps naturally arise by dualizing the multiplication. This immediately implies the abovementioned duality, and the formulae displayed in [5, §4] can also be subsumed under our general approach. By demanding that δ be g-linear, we depart from the usual compatibility condition of a Lie bialgebra, which requires δ to be a derivation. In that case, μ ◦ δ is a derivation of g. For fields of characteristic zero, only nilpotent Lie algebras afford invertible derivations (cf. [7]), so that non-zero Lie bialgebras over such fields never satisfy μ ◦ δ = idg. Let k be a field. Given a finite-dimensional k-vector space V , we consider the k-linear maps

Lie Bialgebra Structures on Lie Algebras of Block Type

This paper considers Lie bialgebra structures on Lie algebras of Block type and proves that all such Lie bialgebras are triangular coboundary.

Equivariant quantizations of symmetric algebras

Let g be a Lie bialgebra and let V be a finite-dimensional g -module. We study deformations of the symmetric algebra of V which are equivariant with respect to an action of the quantized enveloping algebra U h ( g ) , resp. U q ( g ) . We investigate, in particular, such quantizations obtained from the quantization of certain Lie bialgebra structures on the semidirect product of g and V. We classify these structure in the important special case, when g is complex, simple, with quasitriangular Lie bialgebra structure and V is a simple g-module. We then introduce a more general notion, co-Poisson module algebras and their quantizations, to further address the problem and show that many known examples of quantized symmetric algebras can be described in this language.

Quantization of Lie Algebras of Generalized Weyl Type

Recently, in [25], Lie bialgebra structures on Lie algebras of generalized Weyl type were considered, which are shown to be triangular coboundary. In this paper, we quantize these algebras with their Lie bialgebra structures.

Lie bialgebra structures on the Schrödinger–Virasoro Lie algebra

In this paper we shall investigate Lie bialgebra structures on the Schrödinger–Virasoro algebra L. We found out that not all Lie bialgebra structures on the Schrödinger–Virasoro algebra are triangular coboundary, which is different from the related known results of some other Lie algebras related to the Virasoro algebra.

Quantization of Hamiltonian-type Lie algebras

In a previous paper, we classified all Lie bialgebras structures of Hamiltonian type. In this paper, we give an explicit formula for the quantization of Hamiltonian-type Lie algebras.

Compatibility of quantization functors of Lie bialgebras with duality and doubling operations

We study the behavior of the Etingof–Kazhdan quantization functors under the natural duality operations of Lie bialgebras and Hopf algebras. In particular, we prove that these functors are “compatible with duality”, i.e., they commute with the operation of duality followed by replacing the coproduct by its opposite. We then show that any quantization functor with this property also commutes with the operation of taking doubles. As an application, we show that the Etingof–Kazhdan quantizations of some affine Lie superalgebras coincide with their Drinfeld–Jimbo-type quantizations.

Quantization of Lie Bialgebras, Part VI: Quantization of Generalized Kac–Moody Algebras

This paper is a continuation of the series of papers “Quantization of Lie bialgebras (QLB) I-V”. We show that the image of a Kac-Moody Lie bialgebra with the standard quasitriangular structure under the quantization functor defined in QLB-I,II is isomorphic to the Drinfeld-Jimbo quantization of this Lie bialgebra, with the standard quasitriangular structure. This implies that when the quantization parameter is formal, then the category O for the quantized Kac-Moody algebra is equivalent, as a braided tensor category, to the category O over the corresponding classical Kac-Moody algebra, with the tensor category structure defined by a Drinfeld associator. This equivalence is a generalization of the functor constructed previously by G. Lusztig and the second author. In particular, we answer positively a question of Drinfeld whether the characters of irreducible highest weight modules for quantized Kac-Moody algebras are the same as in the classical case. Moreover, our results are valid for the Lie algebra \(\mathfrak{g}(A)\) corresponding to any symmetrizable matrix A (not necessarily with integer entries), which answers another question of Drinfeld. We also prove the Drinfeld-Kohno theorem for the algebra \(\mathfrak{g}(A)\) (it was previously proved by Varchenko using integral formulas for solutions of the KZ equations).